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The vibration of disc cam mechanism
Mech. Much. Theory Vol. 30, No. 5, pp. 695-703, 1995 Copyright © 1995ElsevierScienceLtd 0094-114X(94)00062-X Printed in Great Britain.All rights reserved 0094-114X/95 $9.50+ 0.00
Pergamon
THE VIBRATION
OF DISC CAM
MECHANISM
YUKSEL YILMAZ and HIKMET KOCABAS Faculty of Mechanical Engineering, lstanbul Technical University, G/im/issuyu, 80181 Istanbul, Turkey (Received 21 April 1993; in revised form 27 September 1994; received for publication 11 October 1994)
Abstract--In this paper, the longitudinal vibrations of a follower which is the linear active component of a cam mechanism is studied as if it has a constant cross section. The basic Bernoulli method is applied to solve the partial differential .equation which is supplied by taking the viscous damping factor into consideration and by adding a new term. For the first time, in addition to the viscous damping, in this investigation, the internal damping term has been imposed to the motion of the follower. The resultant integration constants are determined by using the boundary conditions, and the eigen frequencies are calculated. As a result, two different families of eigen frequencies are remained. The results are formed as computer graphics to be easily understandable.
NOMENCLATURE ¢o., o9., ogki.og~--eigen frequencies of system k, k~, k2--spring constants M--reduced mass of the follower (kg) r--damping ratio of the follower Fo--initial force on the cam surface (kg) s, s (~p)--motion curve of the follower (mm) F---cross section of the follower (mm2) L--length of the follower (ram) p--density of the follower (kg/mm 3) c--internal damping ratio E--modulus of elasticity (N/mm 2) tp--rotation angle of cam (rad) mr---roller mass (kg) u (x, t ), u,(x, 1)---displacement of cross section (mm) x--length on the follower with respect to the roller (mm) t--time (s) or--stress within the follower (N/mm2) Ao, Bo, Ai, Bi, A2, B2--constants co--angular speed of cam (rad/s) ct, fl, ?~, ?2--abbreviations given with equations (10) ~---operation force (N) F*--total of the initial tension force and operation force (N) E--unit strain H--maximum displacement of a point on the follower (mm) 2z, 22, 23,2,--abbreviations given with equations (21)
1. I N T R O D U C T I O N T h e essential i n f o r m a t i o n o f k i n e m a t i c s , d y n a m i c s a n d d e s i g n o f c a m m e c h a n i s m s exists extensively in l i t e r a t u r e [1-3]. T h e s e m e c h a n i s m s c a n s u p p l y with n e a r l y all m o t i o n e q u a t i o n s w h e t h e r p r a c t i c a l o r t h e o r e t i c a l a n d c a m realizes all m o t i o n o n c o n d i t i o n t h a t it is fitted in s o m e criterions. U p to d a t e m a n y p a p e r s a n d research articles h a v e b e e n p u b l i s h e d o n this subject. I n these researches, t w o different m a t h e m a t i c a l m o d e l s are p r e f e r r e d for the d y n a m i c a n a l y s i s o f disc c a m m e c h a n i s m s . O n e o f these is the m a s s a n d s p r i n g system which has o n e o r m o r e degrees o f f r e e d o m . T h e o t h e r is a n elastic system w h i c h has infinite degree o f f r e e d o m . T h e basic m a t h e m a t i c a l m o d e l o f a disc c a m m e c h a n i s m is given in Fig. I. W h e r e M is the r e d u c e d mass, k~ a n d k2 a r e s p r i n g ratios. T h e n a t u r a l a n g u l a r f r e q u e n c y in this s y s t e m c a n be w r i t t e n as [4]: '~ -+- k 2
con =
M
695
(1)
696
Yiiksel Yilmaz and Hikmet Kocabas
/ / I /
Fig. 1.
However the mass-spring systems with five degrees of freedom is represented by Janssen [5]. Some simplification is done by neglecting viscous damping factor too [6, 7]. Mathematical model with infinite degrees of freedom is important in the dynamic analysis and synthesis of disc cam mechanism, and this model is fitted to physical model. Only a few authors have taken the infinite degrees of freedom models into consideration to solve the eigen frequencies [8, 9]. In these papers, the authors applied different boundary conditions and neglected the damping ratio by accepting that it is not much effective on the amplitude of vibration [8]. As we agree with them about exhausting of considering this r damping ratio, it will be more appropriate to examine and see the results by considering the problem more general.
2. V I B R A T I O N E Q U A T I O N The mathematical model of the follower of cam mechanism that is considered as an elastic rod with damping is given in Fig. 2. A helical spring with the spring ratio of k is used to supply a continuous contact between the cam and the follower in this system. For this reason, the spring is pressed to apply an F0 initial force. The geometry or manufacturing profile of cam is determined so the shape of the follower can supply the s = s ((p) motion, where the cross section of the follower is F, its length L, density p, the internal damping ratio c and the elasticity modulus E. Here tp is the rotation angle of cam. s = s (tp) can be arbitrarily selected by following some criterions. The mass reduced from the active components of cam mechanism and roller mass m r on x axis is accepted as M. The displacement of cross section, which is rectangular to x axis of the follower, is represented by u (x, t ) as a function o f x and t time. The differential equation is known as [10, 11]: E O2u
p 8x 2
O3u
c F
~2u
p 3x 2&
=
(2)
&2
if c is internal damping ratio, the deformation stress tr is determined by: (~U
(~ 21,/
OX
OxOt
tr = E T - - c
(3)
and the source of this damping factor are internal frictions within the follower material [12]. To solve the differential equation (2), we offer u(x, t) = X(x) (cos ~ot + sin o~t) + (A0 + B0 sin ogt)x
4 I u(x,t) . . . . . .
iM .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
k
.
Fig. 2. Mathematical model (1 support, 2 cam, 3 follower, 4 roller).
(4)
The vibration of disc cam mechanism
697
This is known as Bernoulli method in the literature[13] and it is basic u ( x , t ) = X ( x ) T ( t ) supposition with an additional second term. Here A0 and B0 are constant. If this equation is used in (2). d2X(x)
(~ + ~w) ~
+ to2X(x)
= 0
(5)
d2X(x) . 2... (~ - 3to) ~ -t- to a t x ) = - B o t o Z x
(6)
two traditional differential equations (5), (6) with constant ratio are obtained where E =-- ; P
c fl = - P
(7)
Where c is internal damping ratio, E = d u / d x unit strain and the cross section stress is defined as a = E • c + c • dE/dt.
Also q~ = tot and to is angular speed of cam. The solution of the equation (5) is
(8)
X~ = A, cos 7~x + B~ sin 7~x and the solution of the equation (6) is
(9)
)(2 = A2 cos y2x + B 2 sin V2X - Box where to
to
If X, (x) and X 2 ( x ) are used in equation (4), u, (x, t) = (A, cos 7, x + B, sin 7~x)(cos tot + sin tot) + (A0 + B0 sin tot)x u2(x,t)=(Azcos72x
+ B2sin72x-Box)(costot
+sintot)+(Ao+
Bosintot)x
(I 1) (12)
equations are obtained• A0, A,, A2, B0, B, and Bz ratios can be determined easily from the boundary conditions. The boundary conditions (a) f o r x = 0 ,
t>0; uj (0, t) = u2(0, t) = s (tp)
(13)
(b) for x = L, t > 0; Fo is initial force, F~ operating force, and F* is defined as F* = F0 + Fi
(14)
With these definitions, equation (15) can be written as follows:
-
-
d
dt
~ Ou(x,t).,.=L
=/Z/'
OX
(15)
The equation of motion for cam mechanism is usually selected on periodical of trigonometric function. In case some other motions are selected, these motions can also be converted to the periodical or trigonometrical function by using Fourier transformation. Here, however, the following equation has been chosen to simplify studies. This does not affect the generality of the investigation. The essential motion equation of the cam manufacturing profile is selected as S ( cp) = H (cos tot + sin cot)
(16)
698
Yiiksel Yilmaz and Hikmet Kocabas
By using equation (13) Ai = A 2 = H
(17)
can be written. Here, H is the m a x i m u m displacement o f a point of the follower when it is rigid. By using b o u n d a r y condition equation (15) and equalizing the sinus and cosinus term constants, F*
A0 =
(18)
kL + EF
is obtained. B0, Bt and B 2 terms are quite long and our object is to find the eigen frequencies. W h e n B~ = ~ and B 2 = o{3, for Bi = oo EF?|().3-)-2) 23(k + ro9 - Mo9 2) - ).2(k - to9 - Mo9 2)
tan ~ L =
(19)
and for B 2 :- or3 EFy2 ()-2 --/]-4 ) 22 (k + ro9 - Mo9 2) - 24 (k - ro9 - Mo9 2)
tan 72L =
(20)
equations (18), (19) are determined. Here, 2, = k l + E F
22 = Logr
23 = L ( k - Mo9 2) + E F
)-4 = L ( 092M - k ) - E F
(21)
By using equations (19), (20) and considering equation (10), o9 eigen frequencies can be calculated. Numerical example
With the values o f E = 2 . 1 x l 0 5 N / m m F=5000N/mm, H=200mm, L=400mm,
2, p = 7 . 8 x l 0 - 6 k g / m m 3, M = l k g , c=0.01, r = 0 . 1 , m z = 0 . 3 k g , the values o f four eigen Table 1
Eigen frequencies
~k~
Calculated values
~Ok2
202.8139
COk3
1318.6367
(~k4
2592.7516
3876.6180
tanTt. ,1.00
1.80
0.60 I
/
-0.60
-I.80
I
I
-5.00 0.00
1000,00
?OQQ.O0
~QO.~
qO~.O0
Fig. 3. The calculation of the eigen frequencies,
The vibration o f disc c a m mechanism
699
Table 2 Eigen frequencies
co~'j
to~'2
co~'a
co~'4
Calculated values
202.7203
1318.5513
2592.4315
3875.9067
frequencies are calculated by equation (19) and computer aided iteration method (Table !). One of the graphics of them is given in Fig. 3. With the same values and method, the eigen frequencies for equation (20) are given in Table 2.
901.741
178.026
-183,831 -545,688 -907,546
I.i
901.741
L
"17
i
I "-"-"t
U
178.026
-183.831 -545.688 0002 0,02
,C
t
-907,546
0.000 80.000
400.000 X
O.O0~l
901,741 539,883
tl
178,026
-183.831 -545.688 ~" -907,546
0,100
Fig. 4. The 3 D drawing of the displacements o f cross sections for to = 150.
1443824
U
1443.824
866,222 ~
---"~
866,222
288.62;
./-'-t
288.62I
-288.981
---i
-288,981 -866,582 -1444,I84
-866582 -1444,184 0,000 80.000
400.000
0,00~( 400,001
1443.824
X
U
866.222
288,621 -288,981 -866,582 -1444.184 Fig. 5. The 3 D drawing of the displacements o f cross sections for to = 250.
0050 t
Y/iksel Yilmaz and Hikmet Kocabas
700
1819,758
L!
U
1819.758
1091,728
1091728
363,698
363.698
-364.333
-364,333
-1098,363
-1092,363
-1820,393
-1820,393
0,000 800
160,0
m
X
400.00
3,0
0,00~( 400'000~ X
1819.758
1091,728 363,698 -364.333 -1092.363 0,010 -1820.393
t Fig. 6. The 3D drawing of the displacements of cross sections for 09 = 1300.
The drawings of u~(x, t) and u2(x, t) are useful for understanding and following easily the event, by taking x coordinate on the rod axis and t time as parameters for coki> CO= constant. With the values cot = 150, co2 = 250, co3 = 1300 and CO4~ - 2 0 0 0 , UI(X , t, CO= constant) surfaces can be seen in Figs 4, 5, 6 and 7. In this paper, we have not given the expressions of B0 and BI constants derived by equalizing the constants of cos cot and sin COt in equation (15) while drawing u t (x, t, co = constant) amplitude, because the mathematical expressions of these constants are too long. The numerical values of them are calculated by the use of computer and u~ (x, t, CO= constant) amplitudes are drawn in 3 dimensional form. U
483,787
483,787
290,023
290,023
96.259
96,259
-97.504
-97,504
-291.268
-291,268
-485,032 0.000 80.0
U
-485,032
160.0 240.0
400,00
o.o~
U
483,787 290,023
96.259 -97,504 -291.268 -4R5,032 Fig. 7. The 3D drawing of the displacements of cross sections for o~ = 2000.
0,005 -
t
The vibration of disc cam mechanism
202.81
m
.
175.52
1
14823 120.94 9J,~
6636 ~0,00
M
Fig. 8. The variation of first eigen frequency with mass and internal damping.
UT8.99 1313.54 ~30809
13(]2
65
f29720
f~1.75 ',.~ M
Fig. 9. The variation of second eigen frequency with mass and internal damping.
25~4.18
e~
2591.13 2588.08 i
2585.03 2581,98 2578.93 1.00 2 •
6.40 e.20 tO(X)
M
Fig. 10. The variation of third eigen frequency with mass and internal damping.
701
702
Y/iksel Yilmaz and Hikmet Kocabas 3879.82
tt~
3877.2B
3B74.74
3B72.20
3B69.65
0 C
~67.12 1.00 2 lO.(
Fig. 11. The variation of fourth eigen frequency with mass and internal damping.
At the other side, c d a m p i n g ratio and M mass are i m p o r t a n t for the effect on co eigen frequency. These graphics can be seen in Figs 8, 9, 10 and 11. These four eigen frequencies are recalculated by accepting c as constant, while M is changing. Some numerical sample values are given in Table 3.
CONCLUSIONS A l t h o u g h the following conclusions have been obtained t h r o u g h the interpretation o f a single case studied, these conclusions should be easily applied to m o s t o f the cases. That is to say that the following conclusions are in fact general remarks for the problem involved. 1. ~Oki and ogki*, i = 1, 2, 3, 4 eigen frequencies are differing from each other only in decimal degrees. F o r example, while ~Ok~= 202.8139, C0*~= 202.7203. So that it can be sufficient to calculate only ~ok~s instead of calculating all eigen frequencies. 2. The effect o f c d a m p i n g ratio is low for first, second, third and fourth eigen frequencies while it is high for the higher degree eigen frequencies from Tables 1 and 2. F o r instance while ~k4 = 3876.6180, 09*4 = 3875.9067. 3. The eigen frequencies can be calculated for the various reduced mass M values while c d a m p i n g ratio is constant. In this situation, as the mass becomes small, the values o f eigen frequencies increase. Table 3. Numerical sample values of eigen ~equencies ~kl
202.8130 202.8137 202.8144 202.8136 104.3223 104.3228 104.3219 104.3215 74.1212 74.1218 74.1223 74.1225 66.3597 66.3594 66.3595 66.3597
~k2
1318.5953 1318.7528 1318.9143 1318.9930 1296.2994 1296.4554 1296.6154 1296.6914 1292.5099 1292.6649 1292.8240 1292.9036 1291.7470 1291.9059 1292.0648 1292.1443
~k3
2592.5916 2593.2236 2593.8599 2594.1760 2581.2152 2581.8457 2582.4805 2582.7979 2579.3078 2579.9415 2580.5753 2580.8923 2578.9263 2579.5598 2580.1935 2580.5104
~k4 3876.2603 3877.6833 3879.1110 3879.8230 3868.6448 3870.0704 3871.4925 3872.2058 3867.3732 3868.7979 3870.2231 3870.9359 3867.1189 3868.5434 3869.9684 3870.6811
C 0.000 0.040 0.080 0.100 0.000 0.040 0.080 0.1000 0.000 0.040 0.080 0.100 0.000 0.040 0.080 0.100
n 1.0 1.0 1.0 1.0 4.0 4.0 4.0 4.0 8.0 8.0 8.0 8.0 10.0 10.0 10.0 10.0
The vibration of disc cam mechanism
703
4. u (x, t, co = c o n s t a n t ) a m p l i t u d e surfaces are s m o o t h e r with the smaller values t h a n first eigen frequency while they are b e c o m i n g m o r e c o m p l e x surfaces with the values between the first a n d second eigen frequencies and, second a n d third eigen frequencies, so that the wrinkled surfaces are c r e a t e d out. In conclusion, it has been f o u n d o u t that the effect o f internal d a m p i n g which has been studied first time has been o b s e r v e d after the fourth eigen frequency. This is o b v i o u s l y c o n t r a r y to the c o m m o n u n d e r s t a n d i n g that the d a m p i n g has little effect.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. I I. 12. 13.
REFERENCES F. Y. Chen, Mechanics and Design of Cam Mechanisms. Pergamon Press, New York ([982). H. A. Rothbart, Cams. New York 0956). J. Volmer, Kurvengetriebe. Vcb Verlag Technic, Berlin (1989). J. E. Shigley, Theory of Machines. McGraw-Hill, New York (196I). B. Janssen, Dynamics of Cam Mechanisms (Dynamik der Kurvengetriebe), VDI-Berichte Nr 127, pp. 73 -77 (1966). F. Y. Chen and Polvanich, Trans. ASME, J. Eng. Ind. 97E(3), 769 776. G. K. Matthew, Mech. Mach. Theory 11(4), 247-257 (1976). F. Pasin, Mech. Mach. Theory 9, 239-249 (1974). F. Pasin, Mech. Mach. Theory 18, 151-155 (1983). C. M. Harris and C. E. Crede, Shock and Vibration Handbook. McGraw-Hill, London (1961). W. Seto, Mechanical Vibrations, Schaum's Outline Series. McGraw-Hill, New York (1964). E. Bickel, Die metallischen Werkstoffe des Maschinenbaues. Springer, Berlin (1953). A, Kneschke, Differentialgleichungen und Randwertprobleme, Band H B.G. Taubner Verlaggesellschaft, Leipzig (1961).
Pergamon
THE VIBRATION
OF DISC CAM
MECHANISM
YUKSEL YILMAZ and HIKMET KOCABAS Faculty of Mechanical Engineering, lstanbul Technical University, G/im/issuyu, 80181 Istanbul, Turkey (Received 21 April 1993; in revised form 27 September 1994; received for publication 11 October 1994)
Abstract--In this paper, the longitudinal vibrations of a follower which is the linear active component of a cam mechanism is studied as if it has a constant cross section. The basic Bernoulli method is applied to solve the partial differential .equation which is supplied by taking the viscous damping factor into consideration and by adding a new term. For the first time, in addition to the viscous damping, in this investigation, the internal damping term has been imposed to the motion of the follower. The resultant integration constants are determined by using the boundary conditions, and the eigen frequencies are calculated. As a result, two different families of eigen frequencies are remained. The results are formed as computer graphics to be easily understandable.
NOMENCLATURE ¢o., o9., ogki.og~--eigen frequencies of system k, k~, k2--spring constants M--reduced mass of the follower (kg) r--damping ratio of the follower Fo--initial force on the cam surface (kg) s, s (~p)--motion curve of the follower (mm) F---cross section of the follower (mm2) L--length of the follower (ram) p--density of the follower (kg/mm 3) c--internal damping ratio E--modulus of elasticity (N/mm 2) tp--rotation angle of cam (rad) mr---roller mass (kg) u (x, t ), u,(x, 1)---displacement of cross section (mm) x--length on the follower with respect to the roller (mm) t--time (s) or--stress within the follower (N/mm2) Ao, Bo, Ai, Bi, A2, B2--constants co--angular speed of cam (rad/s) ct, fl, ?~, ?2--abbreviations given with equations (10) ~---operation force (N) F*--total of the initial tension force and operation force (N) E--unit strain H--maximum displacement of a point on the follower (mm) 2z, 22, 23,2,--abbreviations given with equations (21)
1. I N T R O D U C T I O N T h e essential i n f o r m a t i o n o f k i n e m a t i c s , d y n a m i c s a n d d e s i g n o f c a m m e c h a n i s m s exists extensively in l i t e r a t u r e [1-3]. T h e s e m e c h a n i s m s c a n s u p p l y with n e a r l y all m o t i o n e q u a t i o n s w h e t h e r p r a c t i c a l o r t h e o r e t i c a l a n d c a m realizes all m o t i o n o n c o n d i t i o n t h a t it is fitted in s o m e criterions. U p to d a t e m a n y p a p e r s a n d research articles h a v e b e e n p u b l i s h e d o n this subject. I n these researches, t w o different m a t h e m a t i c a l m o d e l s are p r e f e r r e d for the d y n a m i c a n a l y s i s o f disc c a m m e c h a n i s m s . O n e o f these is the m a s s a n d s p r i n g system which has o n e o r m o r e degrees o f f r e e d o m . T h e o t h e r is a n elastic system w h i c h has infinite degree o f f r e e d o m . T h e basic m a t h e m a t i c a l m o d e l o f a disc c a m m e c h a n i s m is given in Fig. I. W h e r e M is the r e d u c e d mass, k~ a n d k2 a r e s p r i n g ratios. T h e n a t u r a l a n g u l a r f r e q u e n c y in this s y s t e m c a n be w r i t t e n as [4]: '~ -+- k 2
con =
M
695
(1)
696
Yiiksel Yilmaz and Hikmet Kocabas
/ / I /
Fig. 1.
However the mass-spring systems with five degrees of freedom is represented by Janssen [5]. Some simplification is done by neglecting viscous damping factor too [6, 7]. Mathematical model with infinite degrees of freedom is important in the dynamic analysis and synthesis of disc cam mechanism, and this model is fitted to physical model. Only a few authors have taken the infinite degrees of freedom models into consideration to solve the eigen frequencies [8, 9]. In these papers, the authors applied different boundary conditions and neglected the damping ratio by accepting that it is not much effective on the amplitude of vibration [8]. As we agree with them about exhausting of considering this r damping ratio, it will be more appropriate to examine and see the results by considering the problem more general.
2. V I B R A T I O N E Q U A T I O N The mathematical model of the follower of cam mechanism that is considered as an elastic rod with damping is given in Fig. 2. A helical spring with the spring ratio of k is used to supply a continuous contact between the cam and the follower in this system. For this reason, the spring is pressed to apply an F0 initial force. The geometry or manufacturing profile of cam is determined so the shape of the follower can supply the s = s ((p) motion, where the cross section of the follower is F, its length L, density p, the internal damping ratio c and the elasticity modulus E. Here tp is the rotation angle of cam. s = s (tp) can be arbitrarily selected by following some criterions. The mass reduced from the active components of cam mechanism and roller mass m r on x axis is accepted as M. The displacement of cross section, which is rectangular to x axis of the follower, is represented by u (x, t ) as a function o f x and t time. The differential equation is known as [10, 11]: E O2u
p 8x 2
O3u
c F
~2u
p 3x 2&
=
(2)
&2
if c is internal damping ratio, the deformation stress tr is determined by: (~U
(~ 21,/
OX
OxOt
tr = E T - - c
(3)
and the source of this damping factor are internal frictions within the follower material [12]. To solve the differential equation (2), we offer u(x, t) = X(x) (cos ~ot + sin o~t) + (A0 + B0 sin ogt)x
4 I u(x,t) . . . . . .
iM .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
k
.
Fig. 2. Mathematical model (1 support, 2 cam, 3 follower, 4 roller).
(4)
The vibration of disc cam mechanism
697
This is known as Bernoulli method in the literature[13] and it is basic u ( x , t ) = X ( x ) T ( t ) supposition with an additional second term. Here A0 and B0 are constant. If this equation is used in (2). d2X(x)
(~ + ~w) ~
+ to2X(x)
= 0
(5)
d2X(x) . 2... (~ - 3to) ~ -t- to a t x ) = - B o t o Z x
(6)
two traditional differential equations (5), (6) with constant ratio are obtained where E =-- ; P
c fl = - P
(7)
Where c is internal damping ratio, E = d u / d x unit strain and the cross section stress is defined as a = E • c + c • dE/dt.
Also q~ = tot and to is angular speed of cam. The solution of the equation (5) is
(8)
X~ = A, cos 7~x + B~ sin 7~x and the solution of the equation (6) is
(9)
)(2 = A2 cos y2x + B 2 sin V2X - Box where to
to
If X, (x) and X 2 ( x ) are used in equation (4), u, (x, t) = (A, cos 7, x + B, sin 7~x)(cos tot + sin tot) + (A0 + B0 sin tot)x u2(x,t)=(Azcos72x
+ B2sin72x-Box)(costot
+sintot)+(Ao+
Bosintot)x
(I 1) (12)
equations are obtained• A0, A,, A2, B0, B, and Bz ratios can be determined easily from the boundary conditions. The boundary conditions (a) f o r x = 0 ,
t>0; uj (0, t) = u2(0, t) = s (tp)
(13)
(b) for x = L, t > 0; Fo is initial force, F~ operating force, and F* is defined as F* = F0 + Fi
(14)
With these definitions, equation (15) can be written as follows:
-
-
d
dt
~ Ou(x,t).,.=L
=/Z/'
OX
(15)
The equation of motion for cam mechanism is usually selected on periodical of trigonometric function. In case some other motions are selected, these motions can also be converted to the periodical or trigonometrical function by using Fourier transformation. Here, however, the following equation has been chosen to simplify studies. This does not affect the generality of the investigation. The essential motion equation of the cam manufacturing profile is selected as S ( cp) = H (cos tot + sin cot)
(16)
698
Yiiksel Yilmaz and Hikmet Kocabas
By using equation (13) Ai = A 2 = H
(17)
can be written. Here, H is the m a x i m u m displacement o f a point of the follower when it is rigid. By using b o u n d a r y condition equation (15) and equalizing the sinus and cosinus term constants, F*
A0 =
(18)
kL + EF
is obtained. B0, Bt and B 2 terms are quite long and our object is to find the eigen frequencies. W h e n B~ = ~ and B 2 = o{3, for Bi = oo EF?|().3-)-2) 23(k + ro9 - Mo9 2) - ).2(k - to9 - Mo9 2)
tan ~ L =
(19)
and for B 2 :- or3 EFy2 ()-2 --/]-4 ) 22 (k + ro9 - Mo9 2) - 24 (k - ro9 - Mo9 2)
tan 72L =
(20)
equations (18), (19) are determined. Here, 2, = k l + E F
22 = Logr
23 = L ( k - Mo9 2) + E F
)-4 = L ( 092M - k ) - E F
(21)
By using equations (19), (20) and considering equation (10), o9 eigen frequencies can be calculated. Numerical example
With the values o f E = 2 . 1 x l 0 5 N / m m F=5000N/mm, H=200mm, L=400mm,
2, p = 7 . 8 x l 0 - 6 k g / m m 3, M = l k g , c=0.01, r = 0 . 1 , m z = 0 . 3 k g , the values o f four eigen Table 1
Eigen frequencies
~k~
Calculated values
~Ok2
202.8139
COk3
1318.6367
(~k4
2592.7516
3876.6180
tanTt. ,1.00
1.80
0.60 I
/
-0.60
-I.80
I
I
-5.00 0.00
1000,00
?OQQ.O0
~QO.~
qO~.O0
Fig. 3. The calculation of the eigen frequencies,
The vibration o f disc c a m mechanism
699
Table 2 Eigen frequencies
co~'j
to~'2
co~'a
co~'4
Calculated values
202.7203
1318.5513
2592.4315
3875.9067
frequencies are calculated by equation (19) and computer aided iteration method (Table !). One of the graphics of them is given in Fig. 3. With the same values and method, the eigen frequencies for equation (20) are given in Table 2.
901.741
178.026
-183,831 -545,688 -907,546
I.i
901.741
L
"17
i
I "-"-"t
U
178.026
-183.831 -545.688 0002 0,02
,C
t
-907,546
0.000 80.000
400.000 X
O.O0~l
901,741 539,883
tl
178,026
-183.831 -545.688 ~" -907,546
0,100
Fig. 4. The 3 D drawing of the displacements o f cross sections for to = 150.
1443824
U
1443.824
866,222 ~
---"~
866,222
288.62;
./-'-t
288.62I
-288.981
---i
-288,981 -866,582 -1444,I84
-866582 -1444,184 0,000 80.000
400.000
0,00~( 400,001
1443.824
X
U
866.222
288,621 -288,981 -866,582 -1444.184 Fig. 5. The 3 D drawing of the displacements o f cross sections for to = 250.
0050 t
Y/iksel Yilmaz and Hikmet Kocabas
700
1819,758
L!
U
1819.758
1091,728
1091728
363,698
363.698
-364.333
-364,333
-1098,363
-1092,363
-1820,393
-1820,393
0,000 800
160,0
m
X
400.00
3,0
0,00~( 400'000~ X
1819.758
1091,728 363,698 -364.333 -1092.363 0,010 -1820.393
t Fig. 6. The 3D drawing of the displacements of cross sections for 09 = 1300.
The drawings of u~(x, t) and u2(x, t) are useful for understanding and following easily the event, by taking x coordinate on the rod axis and t time as parameters for coki> CO= constant. With the values cot = 150, co2 = 250, co3 = 1300 and CO4~ - 2 0 0 0 , UI(X , t, CO= constant) surfaces can be seen in Figs 4, 5, 6 and 7. In this paper, we have not given the expressions of B0 and BI constants derived by equalizing the constants of cos cot and sin COt in equation (15) while drawing u t (x, t, co = constant) amplitude, because the mathematical expressions of these constants are too long. The numerical values of them are calculated by the use of computer and u~ (x, t, CO= constant) amplitudes are drawn in 3 dimensional form. U
483,787
483,787
290,023
290,023
96.259
96,259
-97.504
-97,504
-291.268
-291,268
-485,032 0.000 80.0
U
-485,032
160.0 240.0
400,00
o.o~
U
483,787 290,023
96.259 -97,504 -291.268 -4R5,032 Fig. 7. The 3D drawing of the displacements of cross sections for o~ = 2000.
0,005 -
t
The vibration of disc cam mechanism
202.81
m
.
175.52
1
14823 120.94 9J,~
6636 ~0,00
M
Fig. 8. The variation of first eigen frequency with mass and internal damping.
UT8.99 1313.54 ~30809
13(]2
65
f29720
f~1.75 ',.~ M
Fig. 9. The variation of second eigen frequency with mass and internal damping.
25~4.18
e~
2591.13 2588.08 i
2585.03 2581,98 2578.93 1.00 2 •
6.40 e.20 tO(X)
M
Fig. 10. The variation of third eigen frequency with mass and internal damping.
701
702
Y/iksel Yilmaz and Hikmet Kocabas 3879.82
tt~
3877.2B
3B74.74
3B72.20
3B69.65
0 C
~67.12 1.00 2 lO.(
Fig. 11. The variation of fourth eigen frequency with mass and internal damping.
At the other side, c d a m p i n g ratio and M mass are i m p o r t a n t for the effect on co eigen frequency. These graphics can be seen in Figs 8, 9, 10 and 11. These four eigen frequencies are recalculated by accepting c as constant, while M is changing. Some numerical sample values are given in Table 3.
CONCLUSIONS A l t h o u g h the following conclusions have been obtained t h r o u g h the interpretation o f a single case studied, these conclusions should be easily applied to m o s t o f the cases. That is to say that the following conclusions are in fact general remarks for the problem involved. 1. ~Oki and ogki*, i = 1, 2, 3, 4 eigen frequencies are differing from each other only in decimal degrees. F o r example, while ~Ok~= 202.8139, C0*~= 202.7203. So that it can be sufficient to calculate only ~ok~s instead of calculating all eigen frequencies. 2. The effect o f c d a m p i n g ratio is low for first, second, third and fourth eigen frequencies while it is high for the higher degree eigen frequencies from Tables 1 and 2. F o r instance while ~k4 = 3876.6180, 09*4 = 3875.9067. 3. The eigen frequencies can be calculated for the various reduced mass M values while c d a m p i n g ratio is constant. In this situation, as the mass becomes small, the values o f eigen frequencies increase. Table 3. Numerical sample values of eigen ~equencies ~kl
202.8130 202.8137 202.8144 202.8136 104.3223 104.3228 104.3219 104.3215 74.1212 74.1218 74.1223 74.1225 66.3597 66.3594 66.3595 66.3597
~k2
1318.5953 1318.7528 1318.9143 1318.9930 1296.2994 1296.4554 1296.6154 1296.6914 1292.5099 1292.6649 1292.8240 1292.9036 1291.7470 1291.9059 1292.0648 1292.1443
~k3
2592.5916 2593.2236 2593.8599 2594.1760 2581.2152 2581.8457 2582.4805 2582.7979 2579.3078 2579.9415 2580.5753 2580.8923 2578.9263 2579.5598 2580.1935 2580.5104
~k4 3876.2603 3877.6833 3879.1110 3879.8230 3868.6448 3870.0704 3871.4925 3872.2058 3867.3732 3868.7979 3870.2231 3870.9359 3867.1189 3868.5434 3869.9684 3870.6811
C 0.000 0.040 0.080 0.100 0.000 0.040 0.080 0.1000 0.000 0.040 0.080 0.100 0.000 0.040 0.080 0.100
n 1.0 1.0 1.0 1.0 4.0 4.0 4.0 4.0 8.0 8.0 8.0 8.0 10.0 10.0 10.0 10.0
The vibration of disc cam mechanism
703
4. u (x, t, co = c o n s t a n t ) a m p l i t u d e surfaces are s m o o t h e r with the smaller values t h a n first eigen frequency while they are b e c o m i n g m o r e c o m p l e x surfaces with the values between the first a n d second eigen frequencies and, second a n d third eigen frequencies, so that the wrinkled surfaces are c r e a t e d out. In conclusion, it has been f o u n d o u t that the effect o f internal d a m p i n g which has been studied first time has been o b s e r v e d after the fourth eigen frequency. This is o b v i o u s l y c o n t r a r y to the c o m m o n u n d e r s t a n d i n g that the d a m p i n g has little effect.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. I I. 12. 13.
REFERENCES F. Y. Chen, Mechanics and Design of Cam Mechanisms. Pergamon Press, New York ([982). H. A. Rothbart, Cams. New York 0956). J. Volmer, Kurvengetriebe. Vcb Verlag Technic, Berlin (1989). J. E. Shigley, Theory of Machines. McGraw-Hill, New York (196I). B. Janssen, Dynamics of Cam Mechanisms (Dynamik der Kurvengetriebe), VDI-Berichte Nr 127, pp. 73 -77 (1966). F. Y. Chen and Polvanich, Trans. ASME, J. Eng. Ind. 97E(3), 769 776. G. K. Matthew, Mech. Mach. Theory 11(4), 247-257 (1976). F. Pasin, Mech. Mach. Theory 9, 239-249 (1974). F. Pasin, Mech. Mach. Theory 18, 151-155 (1983). C. M. Harris and C. E. Crede, Shock and Vibration Handbook. McGraw-Hill, London (1961). W. Seto, Mechanical Vibrations, Schaum's Outline Series. McGraw-Hill, New York (1964). E. Bickel, Die metallischen Werkstoffe des Maschinenbaues. Springer, Berlin (1953). A, Kneschke, Differentialgleichungen und Randwertprobleme, Band H B.G. Taubner Verlaggesellschaft, Leipzig (1961).